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79
Designing Programs That Check Their Work
, 1989
"... A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It d ..."
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Cited by 358 (18 self)
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A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable.
Tensor products of modules for a vertex operator algebras and vertex tensor categories
 in: Lie Theory and Geometry, in honor of Bertram Kostant
, 1994
"... In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announ ..."
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Cited by 68 (12 self)
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In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announcement has also appeared [HL1].
Defect Zero pBlocks for Finite SIMPLE GROUPS
 Trans. Amer. Math. Soc
, 1996
"... . We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a pblock with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p\Gammablocks remained unclassified were the alternating groups An . Here we show that these ..."
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Cited by 44 (5 self)
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. We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a pblock with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p\Gammablocks remained unclassified were the alternating groups An . Here we show that these all have a pblock with defect 0 for every prime p 5. This follows from proving the same result for every symmetric group Sn , which in turn follows as a consequence of the tcore partition conjecture, that every nonnegative integer possesses at least one tcore partition, for any t 4. For t 17, we reduce this problem to Lagrange's Theorem that every nonnegative integer can be written as the sum of four squares. The only case with t ! 17, that was not covered in previous work, was the case t = 13. This we prove with a very different argument, by interpreting the generating function for tcore partitions in terms of modular forms, and then controlling the size of the coefficients usin...
Homogeneous Designs and Geometric Lattices
 JOURNAL OF COMBINATORIAL THEORY SERIES A
, 1985
"... During the last 20 years, there has been a great deal of research concerning designs with A = 1 admitting 2transitive groups. The following theorems will be proved in this note; they are fairly simple consequences of the classification ’ of all finite simple groups (see, e.g., [6]). THEOREM 1. Let ..."
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Cited by 43 (4 self)
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During the last 20 years, there has been a great deal of research concerning designs with A = 1 admitting 2transitive groups. The following theorems will be proved in this note; they are fairly simple consequences of the classification ’ of all finite simple groups (see, e.g., [6]). THEOREM 1. Let 9 be a design with,! = 1 admitting an automorphism group 2transitive on points. Then 9 is one of the following designs: (i) PG(d qh (ii) AG(d, q), (iii) The design with u=q3 + 1 and k=q+ 1 associated with PSU(3, q) or *G,(q), (iv) One of two affine planes, having 34 or 36 points [S, p. 2361, or (v) One of two designs having u = 36 and k = 3 ’ [12]. THEOREM 2. Let Y be a finite geometric lattice of rank at least 3 such that Aut 6p is transitive on ordered bases. Then either (i) Y is a truncation of a Boolean lattice or a projective or affine geometry, (ii) 9 is the lattice associated with a Steiner system S(3, 6, 22), S(4, 7, 23), or S(5, 8, 24), or (iii) 6p is the lattice associated with the 65point design for PSU(3, 4). The groups in Theorems 1 and 2 are described in the course of the proof. It would, of course, be desirable to have more elementary proofs of both
Hyperelliptic jacobians without complex multiplication
 Math. Res. Letters
"... The aim of this note is to prove that in positive characteristic p ̸ = 2 the jacobian J(C) = J(Cf) of a hyperelliptic curve C = Cf: y 2 = f(x) has only trivial endomorphisms over an algebraic closure of the ground field ..."
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Cited by 38 (5 self)
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The aim of this note is to prove that in positive characteristic p ̸ = 2 the jacobian J(C) = J(Cf) of a hyperelliptic curve C = Cf: y 2 = f(x) has only trivial endomorphisms over an algebraic closure of the ground field
The status of the classification of the finite simple groups
 Notices Amer. Math. Soc
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Sylow's Theorem in Polynomial Time
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1985
"... Given a set r of permutations of an nset, let G be the group of permutations generated by f. If p is a prime, a Sylow psubgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow psubgroup exists, and that for any two Sylo ..."
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Cited by 24 (8 self)
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Given a set r of permutations of an nset, let G be the group of permutations generated by f. If p is a prime, a Sylow psubgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow psubgroup exists, and that for any two Sylow psubgroups PI, P, of G there is an element go G such that Pz = g‘PI g. We present polynomialtime algorithms that find (generators for) a Sylow psubgroup of G, and that find ge G such that P, = g‘P, g whenever (generators for) two Sylow psubgroups PI, Pz are given. These algorithms involve the classification of all tinite simple groups. 0 1985 Academic Press. Inc. PART I 1.
Hamiltonian paths in Cayley graphs
, 2008
"... The classical question raised by Lovász asks whether every Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log 2 G, such that th ..."
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Cited by 21 (0 self)
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The classical question raised by Lovász asks whether every Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log 2 G, such that the corresponding Cayley graph contains a Hamiltonian cycle. We also present an explicit construction of 3regular Hamiltonian expanders.
Unramified Brauer groups of finite simple groups of Lie type
 Aℓ, Amer. J. Math
"... Abstract. We study the subgroup B0(G) of H 2 (G, Q/Z) consisting of all elements which have trivial restrictions to every Abelian subgroup of G. The group B0(G) serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties V/G where V is a faithful complex linear represe ..."
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Cited by 19 (5 self)
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Abstract. We study the subgroup B0(G) of H 2 (G, Q/Z) consisting of all elements which have trivial restrictions to every Abelian subgroup of G. The group B0(G) serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties V/G where V is a faithful complex linear representation of the group G. We prove that B0(G) is trivial for finite simple groups of Lie type Aℓ.
Primitive permutation groups of odd degree, and an application to finite projective planes
 MR878466 (88b:20007) Zbl 0606.20003 Nick Gill
, 1987
"... One of the most beautiful and important results concerning finite projective planes is the OstromWagner Theorem [26]: such a plane admitting a 2transitive collineation group must be desarguesian. It has long been conjectured that the same conclusion must hold if it is only assumed that there ..."
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Cited by 13 (2 self)
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One of the most beautiful and important results concerning finite projective planes is the OstromWagner Theorem [26]: such a plane admitting a 2transitive collineation group must be desarguesian. It has long been conjectured that the same conclusion must hold if it is only assumed that there